“Flavors of compressive sensing” (3h)

by Simon Foucart (Texas A&M University, USA)

Abstract: Compressive Sensing has attracted a lot of attention in science and engineering, because it revealed the theoretical possibility of acquiring structured high-dimensional objects using much less information than previously expected, and more importantly because it provided practical procedures to do so. The foundations of the field rely on an elegant mathematical theory that has matured over the past few years. The goal of these two lectures is to highlight the main aspects of this theory.

In particular, the following items will be discussed:

• the exact reconstruction of sparse vectors;
• the stable and robust reconstruction of nearly sparse vectors acquired with minor errors;
• the basic reconstruction algorithms (convex programming, greedy algorithms, thresholding-based strategies);
• the restricted isometry property;
• random measurement matrices;
• optimality of uniform recovery results.

“Computational Imaging with Convex and Non-Convex Optimization” (4h)

by Ulugbek Kamilov (Washington University in St. Louis, USA)

Covered topics:

• proximal algorithms in imaging (nonconvex, convex, and strongly convex)
• stochastic optimization (SGD, stochastic extensions of proximal algorithms)
• model-based imaging with plug-and-play priors
• applications to tomographic microscopy

“Optimization strategies for fast inverse problems under sparsity constraints

(with some applications in neuroimaging)” (4h)

by Alexandre Gramfort (Inria, Université Paris-Saclay, France)

In this course AG will:

• Motivate the use of sparse regularizations in the context of neuroscience applications;
• Review some results on coordinate descent methods starting from more well known iterative algorithms such as a proximal or projected gradient descent;
• Cover the dual construction of Lasso-type solvers and explain how it can be used to control optimality, derive accelerations with screening rules and working set methods;
• Present how such methods can be efficiently implemented in Python using Numba or Cython (as done in the Scikit-Learn software).